231 2.3 Functions EXAMPLE 10 Interpreting a Graph Figure 30 shows the relationship between the height of a kite in feet, h1t2, and time in minutes, t. By looking at this graph of the function, we can answer questions about the height of the kite at various times. For example, at time 0 the kite is on the ground. The height then increases, decreases, increases again, and becomes constant. Then the kite returns to the ground. Use the graph to answer the following. (a) What is the maximum height of the kite? When is the maximum height reached? (b) After the kite reaches its maximum height, it begins to fall. What is the kite’s minimum height as shown on the graph between 5 and 15 min? When does this minimum occur? (c) When is the height of the kite increasing? decreasing? constant? (d) Describe the height changes indicated by the graph. t y = h(t) 50 5 10 15 20 25 100 Minutes Height of a Kite Height (in feet) 0 150 200 Figure 30 NOTE To decide whether a function is increasing, decreasing, or constant over an interval, ask, “What does y do as x goes from left to right?” Our definition of increasing, decreasing, and constant function behavior applies to open intervals of the domain, not to individual points. SOLUTION We observe the domain and ask, “What is happening to the y-values as the x-values are getting larger?” Moving from left to right on the graph, we see the following: • Over the open interval 1-∞, -22, the y-values are decreasing. • Over the open interval 1-2, 12, the y-values are increasing. • Over the open interval 11, ∞2, the y-values are constant (and equal to 8). Therefore, the function is decreasing on 1-∞, -22, increasing on 1-2, 12, and constant on 11, ∞2. S Now Try Exercise 91. EXAMPLE 9 Determining Increasing, Decreasing, and Constant Intervals Figure 29 shows the graph of a function. Determine the largest open intervals of the domain over which the function is increasing, decreasing, or constant. x y 0 8 (1, 8) (–2, –1) 3 1 3 –2 –3 –1 Figure 29
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